CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
EXISTENCE OF EQUILIBRIUM IN SINGLE AND DOUBLE PRIVATE
VALUE AUCTIONS
Matthew O. Jackson
CaliforniaInstitute of Technology
JeroenM. Swinkels
WashingtonUniversity in St. Louis
Forthcoming: Econometrica
1 8 9 1
C
A
L
IF
O
R
N
IA
I
N
S
T I T
U T E O
F
T
E
C
H
N
O
L
O
G
Y
Private Value Auctions
MatthewO. Jackson Jeroen M. Swinkels Forthcoming: Econometrica
Abstract
We showexistence of equilibriaindistributional strategies fora wide classof private
value auctions, including the rst general existence result for double auctions. The set
of equilibria is invariant to the tie-breaking rule. The modelincorporates multiple unit
demands, all standard pricing rules, reserve prices, entry costs, and stochastic demand
and supply. Valuations can be correlated and asymmetrically distributed. For double
auctions, we show further that at least one equilibrium involves a positive volume of
trade. The existence proof establishes new connections among existence techniques for
discontinuous Bayesian games.
JEL classi cation numbers: C62,C63,D44,D82
Keywords: Auctions,DoubleAuctions, Equilibrium,Existence, Invariance,Private
Private Value Auctions
MatthewO. Jackson Jeroen M. Swinkels Forthcoming: Econometrica
1 Introduction
1
Auctionsaregenerallymodeledbyallowingplayers tochoosefromcontinuumbidding
spaces. However, players' payo s in auctions are discontinuous at points of tied bids;
which in the face of continuum bidding spaces makes existence of equilibrium diÆcult
to prove. Much of what is known about existence of equilibrium in auctions comes
from exhibiting equilibrium strategies in symmetric settings (for instance in Milgrom
and Weber [23]) or relying on monotonicity arguments (e.g., Athey [1] or Maskin and
Riley [22]). This leaves open the question of existence of equilibria in many auction
settings, such as those where distributions fail to satisfy nice monotonicity properties,
and settingsincludingthe importantclass of doubleauctions.
Sincethe fundamentaldiÆculty inthese proofs revolves aroundthe continuityof the
bid space,one mightwellask why one bothers toimpose suchan assumption. After all,
one could argue that all true bid spaces are in fact discrete. However, continuum bid
spaces are averyuseful approximationasthey simplifythe analysis;allowing,for
exam-ple, one to use calculus to characterize equilibria. Thus, almost all models of auctions
use continuum bid spaces and so it is important to understand when equilibria exist in
such models. Moreover, discrete bid spaces can introduce some pathological equilibria.
Forexample,bothJackson[10]andJackson,Simon,SwinkelsandZame([11],henceforth
JSSZ)showexampleswherethegamewith nitebidspaceshasanequilibrium,whilethe
Wethank LeoSimon, BillZame,MarkSatterthwaite andPhilRenyforhelpful conversations. We
also thankKimBorder,Martin Cripps,JohnNachbar,AndyPostlewaite,LarrySamuelson,Tianxiang
Ye, three anonymous refereesand theeditor for helpful commentsand suggestions. Financial support
from theNationalScience Foundationunder grantSES-9986190andfrom theBoeingCenter forT
ech-nologyandInformationManagementisgratefully acknowledged.
1
This paper supersedes the second part of Jacksonand Swinkels [13]. That paperwassplit: The
results on existenceof equilibria in Bayesiangames with type-dependent sharingwere combined with
Simon andZame[33] tobecomeJackson,Simon, Swinkels,andZame[11]. The resultson existenceof
onone bid, whileanother concentrates onthenext availablebid. This doesnotstrikeus
aswhat wehadinmindwhen discussingequilibrium. Existenceof equilibriumwith
con-tinuumbidding spaceshelps toestablish the existenceof a non-pathologicalequilibrium
in the discrete versions of these auctions. 2;3
Thispaperhasthreemainresultsabout equilibriainauctions: Existence,Invariance,
and Non-Triviality. By existence, of course we mean that the set of equilibria is
non-empty. By Invariance, we mean that the set of equilibria of auctions in our class does
not depend on the precise tie-breaking rule. Invariance actually plays a key role in our
existence proof. And, by non-triviality, we mean that in the set of equilibria there is
always one inwhich trade occurs with positive probability.
Our existence and invariance results apply to auctions that can be single or double
sided, includingsettingswhereplayers areunsureatthetimethatthey bidwhether they
willbenet buyersornetsellers. Thepricingrulecanbequitegeneral,includingboththe
uniformand discriminatory cases,all pay features,entry costs, reserve prices, and many
other variations. Demands and supplies may be for multiple units, and valuations can
beasymmetricallydistributedand followverygeneralcorrelation patterns;neither
inde-pendencenor anyformof aÆliationisassumed. Finally,inadditiontocoveringstandard
tie-breaking rules we also cover more general tie-breaking rules, including allowing the
auctioneer to use information in breaking ties that he is typically not assumed to have,
such as the true values of the players.
In double auctions, showing that an equilibriumexists does not close the issue. The
diÆculty is that double auctions have a degenerate equilibriumin which all buyers bid
0, and all sellers bid v; the upper bound of values. 4
Our second main result establishes
existenceofanon-degenerateequilibriumforeachauctioninourclass; thatis,one where
trade occurs with positiveprobability. So, our results are not vacuous in settings where
there exist degenerateno-trade equilibria. Taken together, these two results providethe
rst generalexistence result of any formfor the double auction setting. 5
2
JSSZalsoarguethatoneshouldbeuncomfortablewiththeequilibriaofdiscretebidauctionsifone
doesnot knowthat they correspond to analogs in the continuum, because absent an existence result
for the continuum case, there is no assurancethat the equilibrium does notdepend on theparticular
discretization chosen. Forexample,itmightbecriticalwhether di erentplayer'savailablebidsoverlap
oraredisjoint.
3
The underlying existence theorems used (either JSSZ orReny), include an upperhemi-continuity
result. Hence,thefact(whichweestablish)thatallequilibriaofthecontinuumgamearetie-freeimplies
theexistenceofalmosttie-freeequilibriain gameswithsmallbidincrements.
4
Such no-tradeequilibria arenotoriouslydiÆcultto overcomein someother settings. Forinstance,
seethediscussionofpositivetradeequilibriainmarketgamesinDubeyandShubik[8]andPeck,Shell,
andSpear [25].
5
Williams [37] shows existence in the particular case that the price is determined by the lowest
winning buybid. Thiscaseiseasier,because(a)thesellersthenhaveasaweaklydominantstrategyto
bidtheirvaluesandhence(b)thegamecanbethoughtofasaonesidedauctionwithahiddenreserve
follows.
First,letusdiscussinvariance. Letanomniscienttie-breakingrule beamapfrombids
and values to distributions over allocations that respects the order of bids. Such maps
include standard tie-breaking rules where ties are broken by some simplerandomization
or xed priorities,but alsoinclude weirderrules, suchas ones wherethe auctioneer uses
other information such as players' types and, for instance, breaks ties eÆciently. Say
that a tie-breakingrule istrade-maximizingif ties between buyers andsellers are broken
in favor of buyers. Then, the invariance result states that if a strategy pro le forms an
equilibrium forone omniscienttie-breaking rule, itremains anequilibriumfor any other
trade-maximizing omniscient tie-breaking rule. Two ideas underlie this result. First,
regardless of the tie-breaking rule, given the continuous distribution over types players
willnotwanttoplayinsuchwayastobeinvolvedintieswithpositiveprobability(except
that abuyerand aseller maybetied provided tradeoccurs between them),asotherwise
some player should bene t from raising or lowering his bids contingent on some types.
So, changing tosome other trade-maximizingtie- breakingrule doesnot change payo s
undertheoriginalstrategypro le. Second,ifaplayerhasanimprovingdeviationrelative
to some strategy pro le and tie-breaking rule, then there is a slight modi cation of the
deviation that is stillimprovingand alsoavoid ties. But then, any improving deviation
atone tie-breakingruleimpliesanimprovingdeviationunderany othertie-breakingrule
as well. Hence, replacing one tie-breaking rule with another does not change the best
replies of the players atthe equilibrium. 6
With the invariance result in hand, there are at least three ways to complete the
proof of existence. First, one can use the main result of JSSZ. They prove existence of
equilibria in Bayesian games in which tie-breaking is allowed tobe \endogenous" in the
sense that it is determined as part of the equilibrium and can depend (in an incentive
compatible way) on the private informationofplayers. A fortiori this is anequilibrium
with omniscient tie-breaking, and hence, by the invarianceresult the strategies involved
arealsoanequilibriumforanytrade-maximizingtie-breakingruleincludinganystandard
one.
The invarianceresultcan alsobe usedtoestablishexistence viathe theorem ofReny
[27] (henceforth Reny). We know of twosuch approaches. In one approach,one uses an
auxiliary resultof Reny's (Proposition3.2), involvinga conditioncalled reciprocal upper
semicontinuity (due to Simon [31], who generalizes Dasgupta and Maskin [7]). This
requires that if one player's payo jumps down at a discontinuity, some other player's
payo jumpsup. Understandardtie-breaking,thisconditionisnotsatis ed(seefootnote
13 of Reny). However, if tie-breaking is chosen to maximize the sum of player payo s,
thenreciprocaluppersemicontinuityissatis ed(as isthestronger conditionofDasgupta
and Maskin). Since such tie-breaking is among the omniscient tie-breaking rules, this
(coupled with invariance)again implies the result.
6
theorem. We thinkthis approachillustratesanimportantway ofapplying Reny'sresult
more generally,and thatithintsatadeeperconnectionbetween Reny andJSSZ.Reny's
basic condition, better-reply-security, requires checking that there are paying deviations
not just relativeto payo s actually available atnon-Nash strategy pro les,but also
rel-ative to all payo s that can be generated as a limit of near-by strategy pro les. The
deviation must pay not only relative to the original strategy pro le, but to small
per-turbations. The rst part of applying Reny thus involves the potentiallydaunting task
of characterizing the set of payo pro les available from such limitsat any given point,
which inparticular can be quitecomplicated at pointsof discontinuity such as ties. We
show, however, that any such payo pro le can be induced by an appropriately chosen
omniscient tie-breaking rule. Then,using the rst idea underlying our invariance result,
that players will play so as toavoid being involved in ties regardless of the tie-breaking
rule, establishes the desired condition. We nd it illuminatingthat all three routes to
existence rely onsome form of omniscienttie-breaking. 7
The lastpieceof the puzzleis toshowthat there are non-degenerate(positivetrade)
equilibria in settingssuch as the double auction. Our approachis to \seed" the auction
with a non-strategic player who is present with small probability, and who then makes
buy or sell bids uniform over the range of values. With this extra player present, it no
longer makessenseforbuyers toalways o er0,orsellerstoalwaysask v;sincesuchbids
never trade, while one could bid more generously, and sometimes pro tably trade with
the non-strategic player. What isless clear is that this impliesan amount of trade that
does not vanish as the probability of the non-strategic player being present goes to 0.
The key is to show that once the non-strategic player is present, competition to trade
with himwillpush the bids of buyers with high values well above0 and sellers with low
values well below v: Essentially then, the extra player sets o a cascade, resulting in a
positiveamountoftradeevenastheprobabilitythattheextraplayerispresentvanishes.
Section 2 discusses related literature. Section 3 presents our private-value auction
setting. Section4.1 shows that the set of equilibria insuch auctionsare invariant tothe
tie-breaking rule. In Section 4.2, we show how invariance impliesexistence of equilibria
for the auctionsinour class. Section5shows thatthese equilibriacan bechosen tohave
non-degenerate tradeinauctionenvironmentswhereno-trade equilibriaareapossibility,
such as doubleauctions. An appendix contains the proofs.
2 Related Literature
The methodof proofwe use forour rst result, ofdemonstratingexistence withan
non-standard tie-breaking rule and then showing that the tie-breaking can be changed to
a standard one, is reminiscent of Maskin and Riley [22]. 8
The strategy of Maskin and
7
This isalso trueofLebrun[18]andMaskinandRiley[22].
8
ThistechniqueisalsousedbyJSSZwhoarguethatifvaluationsareprivateandthedistributionis
eventof atie, aVickrey auctiontakesplace. Forthe auctionsthey consider, the Vickrey
auctionisenoughtoguarantee thatpayo s arepreserved inthelimitasthe nitegridof
bidsgrows ne. Theythenarguethatthesetieswouldneveroccurinequilibriumanyway,
and so the equilibrium is in fact a standard one. So, like them, we show existence in a
game where one does something strange in the event of a tie and then work backward
to show that in many cases of interest this was irrelevant, in our case, by applying our
invariance result. Within the class of private value auctions, we cover a substantially
broader set ofcases than those ofMaskinand Riley,essentially becauseour tie-breaking
methodsallowforpotentiallystranger rules. This allowsus tohandleequilibriumwhere
biddingisnot monotoneintype,and thustocoverawidevariety ofauctionformatsand
informationstructuresnothandledinthepreviousliterature. On theotherhand,Maskin
and Riley'sresultshold for somenon-private value auctions(withaÆliatedtypes) while
our result does not. It is an open question how far techniques like those in this paper
extend beyond private value auctions.
As discussed above, the idea of getting existence in games with augmented message
spaces is also related to Lebrun [18], who looks at rst price single unit private value
auctions in which bids are augmented by messages which turn out to be irrelevant. He
notesthat inaugmenting thegame,andlettingtie-breakingdependonthemessagesent,
he isdoingsomething reminiscentof whatSimonand Zame[32]doingamesofcomplete
information. 9
Thus, in the speci c setting he studies, Lebrun's technique parallels the
one weuse here.
This paper is also related to the literature on existence in games with continuum
type spaces,including, for example,Dasgupta and Maskin[7], Simon[31], and of course
Reny [27]. Reny shows that his condition applies in a multiple unit, private value, pay
your bids auction, a case for which we also prove existence (see his Example 5.2). A
recent working paper by Bresky [4]uses a di erent lineof attack to apply Reny's result
to private value auctions. Neither paper covers the class of settings covered here.
Of course, none of the previous literature has anything to say about the problemof
no-trade equilibriain doubleauctions.
Thuswe move beyond the previous literature infour ways:
1. Weshow abroad invarianceproperty across tie-breaking rules for the equilibriaof
private value auctions.
2. Weshowthattheinvariancepropertyandconsiderationofnon-standardtie-breaking
allowsforthestraightforwardapplicationofeitherJSSZorReny. Thewayinwhich
these resultscanbeleveragedshouldbeinteresting initsown rightandpotentially
of wider applicability.
auctionwithstandardtie-breaking(seetheirExample 3).
9
Theideaofusing messagesto restorelimitcontinuityin a nite approximationsettingshouldalso
more ground than previous results,even in the single auctioncase.
4. Finally,weshowtherealwaysexistsanequilibriuminwhichthereisapositive
prob-ability of trade. This overcomes the problem of no-trade equilibria, and provides
the rst existence result ofany formfor double auctions. 10
Anumberofpapers approachtheexistencequestion insingleunit auctionsby
exam-iningtheassociatedsetofdi erentialequations. Astrengthofthisapproachascompared
to ours is that it allows for interesting comparative static and uniqueness results. Such
anapproachrequires muchmorestructure regardingthe distributionsofvaluationsthan
we require here. For some leading examples, see Milgrom and Weber [23], Lebrun [19],
Bajari[2], and Lizzeri and Persico [17].
Athey [1] considers conditions on games such that a monotone comparative statics
result applies to the best bid of aplayer as his signal varies. Essentially, one imposes a
conditionunderwhich,ifallofi'sopponentsareusinganincreasingstrategy,ihasabest
response in increasing strategies. A strength of Athey's resultis that itdoes not rest on
private values. It does however, require a single dimensionaltype space with something
akin to the monotone likelihood ratio property (MLRP, see Example 1). Recent work
byMcAdams [20] andKazumori [15] extendsthis toa multipledimensionalsettingwith
independent types, inthe former case with a discrete bid space, while in the later, with
a continuum.
Each of the auction papers mentioned above derives the existence of pure strategy
equilibria, while in general we show only the existence of equilibria in distributional
strategies. Thisispartlyduetothe methodsweemploy,butmostly duetothebroadness
of the class of distributions of valuations that we admit. In particular (see Example 2
below) not all auctionsin our setting have such pure strategy equilibria,and so aresult
covering these auctions can at most claim existence of mixed strategy equilibria. In
some settings with positively related valuations, one can start from our existence result
and then independently deduce that all equilibria must be in increasing (and therefore
essentially pure) strategies. We present one such result, generalizing McAdams and
Kazumori for the case of private values. 11
See Reny and Zamir[28] and Krishna[16] for
other interesting recent work onpure strategy equilibria.
10
Since the rst writingof this paper(1999), others havealso looked at existence of equilibrium in
double auctions. Fudenberg, Mobius, and Szeidl [9] show existence of equilibrium in double auctions
withsuÆcientlymanyplayers. PerryandReny[26]addressexistenceindoubleauctionswithadiscrete
bid space. These papers all work in asymmetric aÆliatedor conditionally independent setting, and
deriveincreasingequilibria. OursettinghasneithersymmetrynoraÆliation,butdoesnotruleoutthat
theequilibriafoundinvolvemixing.
11
With independenttypes, these paperscandeal with interdependent values,which we donot. We
would like to be clear that while our basicexistence results for equilibria in distributional strategies
(includingnon-trivialequilibriaindoubleauctions)predateMcAdamsandKazumori's,ourcorollaryon
Webeginbypresenting ourmodelof privatevalue auctions. The modeltreatssingleand
double auctions (as wellas hybrids) in asingle framework.
3.1 The Setting
Let us rst describe the setting in terms of the players, objects, valuations, and
uncer-tainty.
Players
Thereare players N =f1;;ng, alongwith anon-strategic\player" 0,whocan act
as the seller, for example, ina single sided auction.
Objects and Endowments
There is ` < 1 such that each player i 2 N [ f0g has an endowment of e
i 2
f0;1;:::;`g indivisible objects. Objectsare identical. Lete=(e
0 ;e 1 ;:::;e n ) denotethe vector of endowments. Valuations
Eachplayer i2N desires atmost ` objects. Player i'svaluations are represented by
v i =(v i1 ;:::;v i`
). The interpretation isthat ihas marginalvalue v
ih
for anh th
object.
Assumption 1: (PrivateValues)Playerireceivesvalue P
H
h=1 v
ih
fromhavingHobjects.
Forhe i ; we say that v ih is asell value. Forh>e i , we say that v ih isa buy value. Letv =(v 1 ;:::;v n
)bethe vector of valuations of the players.
Types We say that i =(e i ;v i
)is the type of player i,and let =(e;v) denotethe vectorof
types of allplayers. Let
i
f0;1;:::;`gIR `
bethe space ofpossibletypes forplayer
i. 12 Let= 0 n
be the space of typevectors.
Assumption 2: (Compact TypeSpace) is compact.
12
Letvbe such that f0;1;:::;`g [ v;v] .
Uncertainty
The vector 2 is drawn according to a (Borel)probability measure P on. The
marginal of P on
i
isdenoted P
i
; i 2f0;:::;ng. Without lossof generality, take
i to
bethe support of P
i .
Assumption 3: (Imperfect Correlation) P is absolutely continuous with respect to
Q
n
i=0 P
i
; with continuous Radon-Nikodymderivative f.
A3putsnorestrictiononhowe
i andv
i
arerelated,orontherelationshipbetweenany
twovaluesv
ih andv
ih 0
forany given player. Itsimplyimposesthat (e
i ;v i )and(e j ;v j )are
not too dependent. Forinstance, in atwo-player, one-objectauction, if P were uniform
onthediagonalfv j v 11 =v 21 g;thenv 1 andv 2
wouldbeperfectlycorrelatedandP would
not beabsolutelycontinuouswithrespect toP
1 P
2
(theuniformdistributionon[0;1] 2
).
On the other hand, under A3 types can be \almost perfectly correlated" in the sense
that P can place probability one onsome smallneighborhood of the diagonal.
Assumption 4: (Atomless Distributions) P
i (fv
ih
=xg) = 0 for all i 2 N, h 2
f1;:::;`g, and x2[ v;v]:
This assumptionrulesout that particular valuesoccurwith positiveprobability.It is
stronger thanjust assumingthat P
i
isatomless asitrules out, for example,that v
i1 1
while v
i2
is distributeduniformlyon[0;1]. Itallows, however, v
ih =v
ih
0 with probability
one.
WeemphasizethatwehavenotimposedanysortofaÆliationamongdi erentplayers'
values and sothe following exampleis withinour setting. Because this auctiondoesnot
haveanequilibriuminnon-decreasing strategies,it isnot covered by any previouspaper
on existence inauctions.
Example 1 Consider a two-player, private-value, rst-price auction. Values are
uni-formly distributed over the triangle
f(v 1 ;v 2 )jv 1 0;v 2 0;1v 1 +v 2 g:
Here, higher values of v
1
correspond to lower expectations of v
2
, and vice versa. This
auctionhasnonon-decreasingpurestrategyNashequilibrium. Toseewhynot,supposeto
thecontrarythatsuchanequilibriumb
1 ();b
2
()exists. Letus rstarguethatv
2 b 2 (v 2 ) for all v 2
2[0;1). Suppose not, sothat b
2 (v 0 2 )>v 0 2 for some v 0 2 2[0;1). Then,since b 2 is non-decreasing, b 2 (v 2 ) > v 2 for all v 2 2 (v 0 2 ;b 2 (v 0 2
)): But, then, since P(vjv
1 < v 2 ; v 2 2
(v 2 ;b 2 (v 2
)))> 0; it must be that there is a positive probability that at least one of the
players wins with a bid above value, 13
and so would do strictly better to lower his bid
to value. This contradicts equilibrium. Now consider a bid by bidder i when his value
is above 1 ". He knows that the other bidder's value is below ", and thus so are the
other bidder's bids. Thus, i's bids should be no more than ", and so i's bids for values
near 1 are near 0. Therefore, the only possible equilibrium innon-decreasing strategies
is b 1 (v 1 )=b 2 (v 2 )=0 for all(v 1 ;v 2
);which isclearly not anequilibrium.
Assumption 5: (Non-Increasing MarginalValuations)
P(f(e;v)jv
ih v
i;h+1
8i;hg)=1:
A5simplystatesthateachplayer's marginalvaluationsforobjectsarenon-increasing
inthenumberofobjects. Thismakesourlifeeasierintermsofkeepingtrackofincentives.
Inparticular,withincreasingmarginalvaluations,aplayermight ndhimselfsubmitting
the same rst and second bids, and simultaneously wishing he could lower his rst bid
becausehe dislikeswinningone object,but raise hissecondbid becausehe likeswinning
two objects. A related and fuller discussion follows Assumption 9. Whether equilibria
still existin suchsituationsis an open question.
3.2 The Class of Auctions
We consider auctions where each player submits a vector of bids, one for each potential
objectthat they may buyorsell. Of course, the auctionmechanismmay ignore someof
this information,but we allowfor the possibility that itis used.
Bidding and Reserve Prices
For each i 2 f0;:::;ng, a bid b
i 2 IR
`
is a non-increasing vector of ` numbers. We
assume that there existreal numbers b<
b such that the set of allowable bid vectorsfor
iis B i =[b ;b] ` :
Let B denote the set of admissiblebid vectors, B=B
0 B 1 B n .
The requirement that bid vectors be non-increasing is simply a labeling statement,
as bids can always be re-orderedin this manner. It willbeconsistent with howauctions
process bids, inthe sense that higherbids are given priority.
13
Eitherplayer2winsforatleastsomevaluesofv
2 2( v 0 2 ;b 2 (v 0 2
))orelse player1mustbeoutbidding
2evenwhenv
1
islowerthanv
2 .
i i
bids outside ofthat range are weakly dominated. Forexample, indiscriminatoryas well
as uniformprice auctions, a bid b
ih
above v
ih
is weaklydominated by a bidat v
ih
. In an
allpay auction,a bidabove v
ih
isweakly dominatedbybidding0. Inthese settings,one
is making no extra restriction on bidder's behavior in imposing the existence of b . 14
In
settings where there is a highest sensible bid for a buyer, allowing sell bids above that
amount is a convenient way to let a seller \sit out" of the auction, and similarly when
there is a lowest sensible bid for sellers.
Timing
The non-strategic player moves rst, and the remaining players then move
simulta-neously. So, attime 0,b
0
isannounced,then attime 1eachplayeri2N observes
i and submits a bid b i . 15;16;17 Payments
The payment that a player makes or receives depends on the number of objects
boughtorsold. Werequirethatconditionalonthenumberofunitsthatiisallocated,and
conditionalontheendowmentvector,hispaymentvariescontinuouslyasafunctionofthe
vectorofsubmittedbids. Thus, therearecontinuousfunctionst
i
:f0;:::;`g n+2
B!IR ;
such that i's payment is t
i
(h;e;b) in the case where the bid pro le is b; the endowment
vector is e, and he receives h objects. 18
14
Foranexamplein which optimalbuy bidsmaynotbebounded fromabove,consider athird price
auction with three players. Suppose that player3 happens to alwaysbid between 0and 1, and that
player1and2havevaluesthatarealwaysatleast2. Then,eachofplayer1andplayer2wouldlikeany
bidhemakestoalwaysexceedanybidbytheotherplayer. So,optimalbids(atleastinsomescenarios)
are unbounded.
Foranexamplein which optimalbuy bids may notbebounded from below, consider anauction in
whichplayerseachdemandtwounits,andinwhichkunitsareforsale. Thepriceisaconvexcombination
ofthekandk+1sthighestbids,withtheweightonthek-thbeingastrictlyincreasingfunctionofthe
averagebid. Then,abidderwithalowsecondvaluemightwell nditoptimaltosubmita rstbidnear
his rstvalue,but makehissecondbidarbitrarilynegative.
15
Whiletreatedidentically to otherbidvector's,b
0
canbethoughtof as0'sreservepricevector. In
order to have player0 notparticipate in theauction at all (forinstance in a double auction) we can
simplyset e
0
=0andb
0h
=bforallh,inwhichcaseallof0'sbidsarenon-competitive.
16
Secretreservepricesarehandledbyhavingplayer1havetheonlypositiveendowment,sothatplayer
1isthesellerandhisbidisthesecretreserve.
17
Note that we are considering the game having xed b
0
, and not the game in which b
0
is chosen
strategically. ItfollowsfromTheorem2ofJSSZthatthesetofequilibriaofthegamede nedbyb
0 with
omniscienttie-breakingisupperhemi-continuousinb
0
:ByTheorem 9below,everysuchequilibriumis
an equilibrium under standardtie-breaking. Hence,theset of Nashequilibria of thegameinduced by
b
0
is upperhemi-continuous. Itfollowsthat thereis alsoan equilibriumof thegame in which buyer0
choosesb
0
accordingtosomeobjective.
18
Thewayin whichwehavede ned t includes aspeci cationof paymentsforb and hwherein fact
present in an auction setting. This is because t
i
(h;e;b) only says what i would pay if i
were to receive h objects. A change in bids can still change how many objects i gets,
say from h toh+1, and thus can still lead to adiscontinuous change in payments. For
example, in a rst price auction, t
i
(1;0;b) = b
i and t
i
(0;0;b) = 0; both of which are
clearly continuouseven thoughthepaymentasafunction ofbids accountingfor tiesand
changesin the numberof objectsreceived isnot continuous.
Typically (but not always, as illustrated by Example (4) below), t
i
(h;e;b) will have
the same sign asi's net trade, h e
i .
19
Payoffs
Players evaluate the outcome of the auction via von Neumann-Morgenstern utility
functions. This allows for risk-averse, risk-loving, or any of a variety of other sorts of
preferences.
Assumption 6: (Expected Utility) Player i 2 N has a von Neumann-Morgenstern
utilityfunctionU
i
overhernet payo . U
i
iscontinuous,strictlyincreasing andhasa rst
derivative that is bounded away from 0and 1.
A6 impliesthat there exists <1 such that U 0
(x)=U 0
(y)< for allx and y:
So, a player's utility when receiving h objects in the nal allocation when the bid
vector is b and the endowment is e isdescribed by
U i 0 @ 0 @ X h 0 h v ih 0 1 A t i (h;e;b) 1 A ; where U i
is acontinuous vonNeumann-Morgenstern utility function.
3.3 Examples and a Preview of Our Results on a Narrower
Class of Auctions
Thegeneraldevelopmentthatfollowsisinvolvedgiventhebreadthoftheclassofauctions
handled andtheattentionpaidtoallocationsandtie-breakingrules. Thus, weo ersome
19
An importantpointaboutthewayinwhichwehaveformulatedthepaymentruleisthatplayeri's
paymentcan depend on his ownallocation but doesnot further depend on other players'allocations,
such as which players other than i won. Without this assumption, an entirely new and tricky set of
discontinuitiesarise. Forexample,eventhoughplayeri mighthappennottobeinvolvedinatie,small
changes in his bid mighta ect which of two opponentswins a tie (rememberthat anomniscient
tie-breakingruleallowsforthispossibility). Ifthisresultsinachangeinthepaymentruleifaces,thenhis
fora narrower class of auctions. This class stillincludesmost standard auction formats.
Although our statements here should be clear, we refer the reader tothe subsequent
sectionsfor the formaland more general statementsof our results.
We emphasize that in all of the examples that follow there is no assumption about
symmetry ofthedistributionofthe players' endowments, valuations,orutilityfunctions.
(1) A standard rst price single unit auction.
In terms of our de nitions and notationthis is expressed as follows. There is one
objectsold by player0,and so` =1 andthe distributionoverendowmentsis such
that Pr(fe=(1;0;:::;0)g)=1. The paymentsare suchthatanagentpayshisbid
if he wins an object (t
i
(1;e;b) = b
i1
) and nothing otherwise (t
i (0;e i ;b) =0). The reserve price is b 01 =0. Let C fi2N jb i1 b j1
for all j 2 N [f0gg(the set of
players who submitted the highestbid). Then, the allocationrule gives the object
to playeri 2N with probability 1=#C if i is in C and 0 otherwise. If C is empty
(so that noplayerother than 0 bids at least 0),then player0 retainsthe object.
Note thatbecausethe sellersubmits abid at0;a negative bid by any otherplayer
neverwins. Hence,suchbidsare simplyawayforplayerstoexpressthatthey have
no interest inwinning.
(2) A standard rst price single unit auctionwith a known reserve price, r0.
This isthe same as Example (1), except that player0 sets a reserve priceb
01 =r.
(3) A singleunit Vickrey (second price) auction.
Thisisasin(1)or(2),exceptthatthepaymentruleforawinningbidderchangesto
t i (1;e;b) =b 2 , where b 2
is the second highest bid submitted (including the reserve
price b
01 =r).
(4) An unfairauction.
This is the same as any of the above examples except that some players pay only
some fraction of the payments indicated abovewhen they win while otherplayers'
payments are unchanged.
(5) An auction with entrycosts (and areserve price).
Let c0bethe entry cost incurred by a playerwishing to makea bid, where the
decision of whether and how to bid is made without knowing other players' entry
decisions. This is handled in our modelas follows. The settingis as in(1), (2), or
(3), except that t i (0;e;b)=cminfb i1 +1;1gand t i (1;e;b)=cminfb i1 +1;1g+b i1
for the rst priceversion (witht
i
(1;e;b)=cminfb
i1
+1;1g+b 2
gforasecondprice
i
assumptions. E ectively, sending a bid of 1 means that the player stays out of
the auctionand doesnot pay the cost c, whereas sendingany bid b
i
0 incursthe
cost c of participating. The remainingbids between -1 and 0 are bids that would
neverbeused inequilibriumsince they cannotwin anobject(given areserveprice
of r 0) and yet would incur some bidding cost. Thus, the presence of the bids
that lie above -1and below0 isjust a technical device in this example.
(6) An all pay auction(and various implementations of the warof attrition).
A rst priceallpayauctionisthesameasin(1)exceptthatt
i (0;e;b)=t i (1;e;b)= b i
. In the standard war of attritionthe winnerpays the second highestbid and so
t i (1;e;b) =b 2 asin (2), while t i (0;e;b)=b i .
(7) A rst-price procurement auction.
Here `=1and player0 has e
0
=0:Thus, setting b
01
>0 representsthe maximum
amount that 0 will pay for an object. Each player i > 0 has e
i
= 1. The lowest
bidder amongi>0sellsanobject toplayer0provided the bid isnomorethan b
01
(with ties among players i > 0 broken in any way). The payment if an object is
sold by ito 0is t
i
(0;e;b)= b
i1
. That is,the buyerpays b
i1
tothe winning seller.
Otherwise payments are 0. The obvious variation leads to a second-price version
of a procurementauction. 20
(8) A multi-unitdiscriminatory(pay-your-bid)auction
Take ` > 1 and Pr(fe = (`;0;:::;0)g) = 1. The top ` bids are declared winners,
and payments are t
i (h;e;b)= P h w=1 b iw .
(9) A multi-unituniform price auction
As in (8), except that winning players pay the `+1-st highest bid for each unit
they acquire, sot i (h;e;b)=hb `+1 , where b `+1
is the `+1-th highest bid. 21;22
(10) A standard doubleauction.
Players 1 through n
b
are potential buyers having e
i = 0. Players n b +1 through n = n b +n s
are potential sellers having e
i
=1. Ties between a buyer and a seller
are broken in favor of trade, and ties among buyers or among sellers are broken
randomly. Letp= b 0 +b 00 =2;whereb 0 isthen s
-thhighestbid, andb 00 the n s + 1-th. Then,t i (0;e;b) = pe i ; while t i (1;e;b)=p(1 e i ): 20
Extensionstomulti-unitprocurementauctionsarealsoeasilyhandled.
21
Therearemanyvariationsonwaystoselectthepricepaid,includingsomethatensurethataplayer's
losing bid does not end up setting the price he pays for his winning bids (such asthat suggested by
Vickrey [36]). Theparticularsofhowthepriceischosenand evenwhetherit di ersacrossplayerswill
notmatter,asourtheoremwill applyinanycase.
22
Examples(8)and(9)covertheclassofprivatevalueauctionsexaminedinSwinkels([34],[35]),and
Players 0 through n draw a realization of (e
i ;v
i
), and submit bid vectors.
Ob-jects are allocated to the P
n
i=0 e
i
highest players, with tie-breaking as in (9). Let
t
i
(h;e;b)=p(h e
i
),wherepisaweaklyincreasingandcontinuousfunctionofthe
( P n i=0 e i )-th and ( P n i=0 e i
)+1-th highest bids. Note that players may turn out to
bebuyers orsellers,evenforagivenrealizationoftheirown typevector,depending
on howtheir bid vector compares to those of other players.
(12) A doublediscriminatory auction.
This is the same as (11), except for the payments. If a player ends up as a net
buyerwith h objects, he pays P h h 0 =e i +1 b ih
0. If iends up as anet seller,he receives
P e i h 0 =h+1 b ih
0:The auctioneer (player0) pockets the di erence.
Theorem 2 Each of the auctions described above has an equilibrium in distributional
strategies which have support in the closure of the set of undominated strategies.
In one-sided auctions (or more generally, any auction where there is a non-strategic
sellerwithareservebelowv); theequilibriumabovewillautomaticallyhavetrade. When
there isno non-strategicseller, this isless clear. Forexample in adouble auction,there
may exist degenerate equilibria where all sellers bid at the top of the support of values
and buyers bid atthe bottom.
Existence of equilibria with a positive probability of trade is guaranteed with two
additionalassumptions. First, werequire that changingone player's type doesnot alter
the support of types for another. In particular, we assume that the Radon-Nikodym
derivative of P with respect to
i P
i
is always positive. Second, we assume that there is
some competition for gains from trade. It is enough to have the support of buyer and
sellers'valuations overlap for allh and to have either atleast two buyers oratleast two
sellers. A weakercondition is described inSection 5.
Theorem 3 Under the above-mentioned assumptions, each of the auctions described
abovehas an equilibrium thathas support in the closureof the setof undominated
strate-gies and has a positive probability of trade.
3.4 TheGeneral Classof Auctions: Allocations andTie-Breaking
Rules
Wenowreturntothe formalde nitionsofallocationsandtie-breaking,whichare needed
in the fullstatement ofour results and tocomplete adescription ofthe class of auctions
An allocation is a vector a 2 f0;`g n+1
A. The component a
i
is the number of
objects that are allocatedto playeri.
Consistent Allocations
An allocationis consistent with vectors of endowments and bids (e;b) if
n X i=0 a i = n X i=0 e i : and fb jh 0 >b ih and a i hg)a j h 0 :
The rst condition is simply a balance condition, requiring that all objects be
ac-counted for. Note that this allows for the possible interpretation that objects that are
allocated to the 0 player might be \unsold," for instance in the case where player 0 is
the onlyseller in anauction.
The second condition simply says that if i receives at least h 1 objects and j's
h 0
-thbid exceeds i'sh th
,then j must get atleast h 0
objects. Thus, higherbids are given
priority overlowerbids in allocatingobjects. 23
Let C(e;b) A denote the set of consistent allocations given endowment and bid
vectors (e;b).
Ties
Say that there is a tie given (e;b) if there exista and a 0
in C(e;b) such that a 6=a 0
.
Say the tieis atb if #fi;hjb ih >b g < n X i=0 e i ; and #fi;hjb ih b g > n X i=0 e i :
So, inthe event ofa tieatb
,allbids aboveb
are lled, but thereis somediscretion
in to whom to allocate objects at b
. Thus, for instance, it is not a tie if there are two
23
In some auctions, some players enjoy a special status. For example, some of the PCS auctions
subsidizedbidsbyminorityowned rms(seeCramton[5]). Onewayofimplementingthiswould beto
declarethe minority rmawinnerifitsbid isatleast, say,2/3of thehighestbid. Weinsteadinclude
asymmetries byinsisting onthehighestbid winning,but allowingpaymentrules todi er, sothat, for
whichare the next-highest. Here bidder twohas the \tied" bids, but willalways get one
object inany allocation.
Tie-Breaking Rules
As discussed in the introduction, we prove existence for a very wide class of
tie-breaking rules, including some fairly strange ones. In particular, we allow for the
pos-sibility that the auctioneer uses more information than just bids and endowments in
determining allocations.
An omniscient tie-breaking rule is a (measurable) function o : B ! (A) such
that o(e;v;b) places probability one on the set of consistent allocations C(e;b). We let
o(e;v;b)[a] denote the probability of allocation a under o at (e;v;b); and o
i
(e;v;b)[h]
denotethe probability that i is allocatedh objectsunder o at (e;v;b). 24
Given the requirement of consistency, o only has any discretion where there are tied
bids, and hencethe term \tie-breakingrule" is appropriate.
Let standard tie breaking be the particular tie-breaking rule which is de ned as
fol-lows. Consider a tie at b
: First, allocate an object to each bid that is strictly above
b
. Next, allocate an object with equal probability to each player who has an un lled
buy bid at b
: Repeat until all objects are gone, or until there are no un lled buy bids
at b
. At this point, iteratively allocate any remainingobjects one at a time with equal
probabilityto those players who have an un lledsellbid atb
:
Thetwokeyaspectsofstandardtie-breakingare rstthattheruleistrade-maximizing,
and secondthat a bidder's chanceof winning anh th
object atb
doesnot depend on i's
other bids. This wouldbefalse,if, forexample,onesimplyrandomlyassignedremaining
objects equiprobably over all bids at b
; as then an h+1 st
bid of b
would increase the
chance that ireceives object h:
While we were led to consider omniscient tie-breaking rules for their use as an
in-termediate step in the proof of existence, it alsostrikes us that there may be situations
in which tie-breaking that depends on more than just bids might be appropriate. For
example, thegovernmentmay have objectivesbeyond thoseof revenues that wouldpush
them to favorone playerover another inthe event ofa tie. In this value setting, wewill
24
Imaginetheauctioneerhadaccesstosomeotherinformation,possiblycorrelatedwithplayertype,
but unobservable to the players at the time they bid. Allowing the auctioneer to also condition on
this information in breaking ties would not expand the set of equilibria beyond those achieved with
omniscienttie-breaking: fromthepointofviewoftheplayers,thisisequivalenttotheauctioneersimply
turnsout tobeirrelevant. It isanopen question whether meaningfulties occurin other
settings, and whether the possibility of favoritism etc., would have an interesting e ect
inthose settings.
Competitive Ties and Trade Maximization
It should be noted that not all ties are the same. On the one hand is a situation in
whichtwobuyers aretied atagiven bid, andonlyone ofthem receives anobject. As we
will show, at least one player will always have an incentive to deviate in this situation.
Consider onthe other hand, asituationin which a singlebuyer and a single seller make
a tied bid, but the object istransferred from buyertoseller ata priceunder which both
are happy to trade. Here, C(e;b) has more than one element, since it is also consistent
for tradenot to occur. But,since the object isactually transferred, thereis noincentive
for either playerto change their bid.
Letussay that (e;b) has acompetitivetieifthere exists apsuchthat the numberof
buy bids that are greater than or equalto p is not the same as the number of sell bids
that are less than or equaltop.
Itturnsout thatwhileequilibriumconditionswillnaturallyrule outcompetitiveties,
non-competitive ties may occur in equilibrium. In particular, itis possible that a buyer
and seller have a tied bid. As long as trade always occurs in this situation, this is not
inconsistent with equilibrium. This iscaptured inthe following condition.
A tie-breaking rule o is trade-maximizing at (v;e;b) if the rule does not specify an
allocationinwhichoneplayerhasanun lledbuybidatbandanotherhasanunaccepted
sellbidatb. 25
Atie-breakingruleoistrade-maximizingifitistrade-maximizingatevery
(v;e;b).
We willbeworking with distributionalstrategies (see Section 3.5for details). Given
a probability measure m on B, say that the rule o is e ectively trade-maximizing if
it is trade-maximizingona set of (v;e;b) havingmeasure 1under m. So, given the way
in which types are drawn and players randomize over bids, the probabilitythat there is
a non-trade maximizingtie iszero.
Thenext example illustratesthe importanceofthe trade-maximizationinour
invari-ance result.
25
Thisincludesplayer0:So,forexample,itisonlynecessarytomeetareserveprice,notstrictlybeat
[3;4];andasellerwithavaluationforasingleunitforsaleandvalueuniformlydrawnfrom
[0;1]; where values are independent across players. The price is the midpoint between
the bids.
If in the event of a tie between a buy and sell bid the auction mechanism speci es
that tradeshouldoccur,then itisanequilibriumforbothplayers tobid 2,andfor trade
toalwaysoccurifboth players bid2. Thisrule ise ectivelytrademaximizing. Ifinstead
the auctionmechanismspeci esthatintheeventofatie, tradeoccurswithaprobability
<1, thenthis isno longeranequilibrium. Now abuyerwould bene tby slightlyraise
his bid, or a seller would bene t fromslightly lowering her bid. In fact,now there is no
longer any equilibriumin which trade always occurs. To see this, suppose the contrary.
Then, almost every bid by the buyer must exceed almost every bid by the seller. But
then, a bid near the bottom of the support of buyer's bids wins almost always, and so
doesstrictlybetter thanahigherbid. Thus, the buyermust bemakingthe same bidb
B ,
regardless of valuation. Similarly,the seller must bemaking the same bid b
S
, regardless
of value. Suppose that b
S <b
B
: Then, a seller can raise his bid and still almost always
sell at a better price, a contradiction. Hence b
S = b
B
= p for some p: But this is a
contradiction sincethesuppositionisthattradeoccurswithprobability<1atatie.
Payments and Bids
Weneedtosaysomethingabouthowbidsdeterminepayments. First,werequirethat
foranygivenallocation,aplayerwhoisanetbuyerisweaklybettero tohavesubmitted
lowerbids, and aplayerwho isa net seller isweakly better o to have submitted higher
bids. Of course, this is holding the allocation constant. Such a change in bid may well
result in the loss ofa pro table trade. Second, wewillrequire that if one is anet seller,
one's buy bids do not matter,and viceversa.
Assumption 7: (Monotonicity)Foranyi,and he
i ,t i (h;e;b) isnon-decreasing inb ih 0 for h 0 > e i and constant inb ih for h 0 e i ; and if h <e i then t i (h;e;b) is non-increasing in b ih 0 for h 0 e i ; and constant inb ih 0 for h 0 >e i :
Note that the condition does not impose any requirements about how a player's
payment depends onthe bids of others.
Having buy payments be independent of sell bids, and vice versa, is useful in our
weak dominationarguments (for instance Lemma 5 below), and also inestablishing the
existence of positivetrade (Theorem 15).
marginal payment when one's h th
bid is involved in atie is a function only of e
i
;h; and
b
ih
: One's other bids, and the details of how many other players one is tied with, and
what their other bids were, are irrelevant.
Assumption 8: (Known MarginalTransfers at Ties) Forall i,h, and e
i ,there is p ihe i : [b ;
b] ! IR such that if (h;e;b) is such that there is a tieat b
ih , then t i (h;e;b) t i (h 1;e;b)=p ihe i (b ih ).
Thisconditionisgenerallysatis edandeasytocheck. Forinstance,fordiscriminatory
auctions, uniform price auctions, and all double auctions (where the price is set in the
range of marketclearingprices forthe submitted demand and supply curves), p
ih (b ih )= b ih . 26;27
It is alsosatis ed for anall pay auction,where the di erence in payments does
not depend on whether the player gets an object and so p
ih (b
ih
)= 0. It is not satis ed
for a third price auction for a single unit, since then, even if the rst two bids are tied,
the price paid may vary depending on the third bid.
Ourinvarianceresultsdonot holdwhen marginaltransfers mightbedecreasing inh:
This e ectivelyinducesavolumediscount,whichcreatesmuchthe samephenomenon as
an upward sloping demand curve: at some bid vectors where the player's two bids are
tied, the player willbe unhappy to win asingle object,but happy to win two.
Assumption 9: (MonotonicMarginalPayments)Foreachi;e
i ;p ihe i (b)isnon-decreasing in h:
This assumption istrivially satis edwhere p
ihe i (b)=b orp ihe i (b)=0.
To see an example where in the absence of such a condition one might get a rather
odd equilibrium and how this mightdepend onthe tie-breaking rule, consider a case in
which each ofthree players has marginalvalue 4for 2objects. Half the time,one object
is available, and half the time, four. Assume that payment rules are such that, when
there isatieat abid ofb
;aplayerpays 6fora rstunit and 1for asecond. Then, itis
an equilibrium for allthree players to bid (b
;b
) always, aslong as tiebreaking is that
when there is a tie and a single object, each player receives the object one third of the
time, while when there is a tie and four objects, each player receives two objects with
probability two thirds, and no object with probability one third. Then, by submitting
(b ;b ); a player earns 1 2 1 3 (4 6)+ 2 3 (8 6) = 1 3 : 26
Notethatinadoubleauctionwhereaplayerhasasingleunittosellandb
i isinatie,t i (0;1;b i )= b i andt i (1;1;b i )=0:So,p i1 =0 ( b i )=b i : 27
This also holdsfor Vickrey auctions wherethe priceisthe highestbidamong other players. Note
thatinthecaseofatie,sincethatrequiresthattheremustbesomediscretionintheawardingofobjects,
rulesofconsistentallocations,thismusthavehimalwaysbeingallocatedtheobjectwhen
there is only one, and paying at least 6. In the most favorable case, it always has him
also win two objects whenever there are four available. Hence, he earns at most
1 2 (4 6)+ 1 2 (8 6)=0
from this deviation. Lowering just the second bid results in sometimes winning a rst
objectataloss, and neverwinningasecondobject. Lowering both bidsresults inpayo
0. The problemhere is that the player would e ectively like toraise his second bid and
lowerhis rst, which is infeasible.
3.5 Strategies and Equilibrium
Given the de nitions from the previous subsections, an auction is a speci cation of
(P;o;t;b
0
). That is, an auction consists of a probability measure, a tie-breaking rule,
a payment rule,and areservepricevector. Inwhatfollows,insome casesit willbeclear
that these are given and we omit mention of them.
We now turn to formal de nitions of the game induced by the auction in terms of
strategies and equilibrium.
Wewritei'sexpectedutilitygivena(possiblyomniscient)tie-breakingruleo,payment
rule t, bid pro le b, valuationvector v, and endowment pro lee as
u i (o;t;b;e;v)= ` X a i =0 o(e;v;b)[a i ]U i 0 @ 0 @ X ha i v ih 1 A t i (a i ;e;b) 1 A : (1) Strategies
A(distributional)strategyforplayeriisa(Borel)probabilitymeasure m
i onB
i
i
that has a marginaldistributionof P
i on
i .
See Milgrom and Weber [24] for discussion of distributional strategies.
Given apro le ofdistributional strategies m
1
;:::;m
n
, playeri'sexpected payo can
bewritten as: i (m;P;o;t;b 0 )= Z u i (o;t;b;e;v)dm 1 (b 1 je 1 ;v 1 ):::dm n (b n je n ;v n )dP(e;v):
When someoftheargumentsare xed,weomitthemfromthe notation,andforinstance
Apro leofdistributionalstrategiesm
1
;:::;m
n
isanequilibriumforauction(P;o;t;b
0 ) if i (m;P;o;t;b 0 ) i (m i ; c m i ;P;o;t;b 0 )
for all i and strategies c
m
i .
Weak Dominance
Aswewish toproveexistenceof equilibriathatsatisfyare nement thatwillruleout
some trivial equilibria, we establish that players use strategies in the closure of the set
of undominated strategies. The formalde nitions are as follows.
Say that bid vector b
i is weakly dominated ate i ;v i by b 0 i if u i (o;t;b i ;b 0 i ;e;v)u i (o;t;b i ;b i ;e;v); for any e i ;v i ;b i
, with strict inequality at least one such pro le, where o is standard
tie-breaking. Wesay thatb
i
isundominatedate
i ;v
i
if itisnot weaklydominatedbyany
other bid.
Note that we include the bid of player 0 in this de nition, which is not completely
standard, as,atthetimethataplayersubmitshisbid,healreadyknowsb
0
:Thisprovides
for a stronger result and actually simpli esthe proofs.
Itisworthdiscussingwhyourde nitionofweakdominanceisrelativetostandard
tie-breaking. Withnon-standardtie-breaking, somepretty odd behaviorsare undominated,
especially in the multiple unit demand case. Consider an example where two players
each value two units. The auction rule is that allobjects are sold at the lowest winning
bid. Most of the time, 2 objects are available. Occasionally, there is only 1. Finally,
player 2 always submits two bids of 3. The tie breaking rule, for whatever reason, is
that if there is a tie at3, and player 1's rst bid is 6, then both objects goto player 1.
If player 1's rst bid is anything else, the second object is allocated at random. Then,
whenplayer2has valuevector(5;4);itisundominatedforhimtobid(6;3);even though
his rst bid is higher than his rst value. Under standard tie breaking, of course, one's
rst bid is irrelevant to the probability that one's second bid is lled if one is involved
in a tie, and such a bid vector is indeed weakly dominated. Although this example is
clearly arti cial, either explicitly or implicitly, either approach to existence, via either
JSSZ orReny,requiresustoadmitthis typeofthingasapossibility. Thustoreallyhave
the appropriate bite on weakly dominated strategies, we rule them out under standard
tie-breaking rules.
Itiswell-known thatexistence ofequilibriumingames withcontinuousactionspaces
intheauctionsetting,wecannotmeaningfullyrequirethatadistributionalstrategyputs
weight zero on weakly dominated bids. It is, however, coherent to require that the
dis-tributional strategy put probability 1onthe closure of the set of non-weakly dominated
bids.
To formalizethis, letW 0 i i B i be the set of e i ;v i ;b i such thatb i isundominated for i given e i ;v i :Let W i bethe closureof W 0 i .
Assumption 10: (UndominatedStrategies). Foreach player i2N; there is a
measur-able map ! i : i B i ! i B i
such that for each (e
i ;v i ;b i ); ! i (e i ;v i ;b i ) = (e i ;v i ;b 0 i ) whereb 0 i =b i if(e i ;v i ;b i )2W i ;andb 0 i weaklydominatesb i given(e i ;v i )if(e i ;v i ;b i )2= W i .
A10statesthatonecan,inameasurableway,identifybidssothatwhenever(e
i ;v i ;b i )2= W i then b i
isreplacedby abid thatweakly dominatesitand resultsinanelementofW
i :
Of course, for this to be satis ed, one needs to know that W
i is in fact non-empty relative to each (e i ;v i
): For general games with continuum action spaces, this need not
beso. Consider agamewithactionspace[0;1];andpayo sequaltoactionforallactions
less than 1, but equal to -1 for action 1. Then, all actions are weakly (in fact strictly)
dominated. The question in the auction setting is whether similar things might arise,
especiallyonceoneconsiderswhathappenstopayo satties(whereitiseasytoconstruct
bid vectors relative towhichthere is nobest response).
We have not found a sensible auction-like example where A10 fails. For example, in
a rst price auction(or a discriminatory multiple unit auction), any bid less than value
is undominated: any higher bid may simply result in bidding more in situations where
one might already have won, while any lower bid may result in the loss of a pro table
purchase. W thus includes all bids in which one bids at or below value. Similarly, in
a second price auction, a bid equal to value is not weakly dominated. And, in either
case, replacing buy bids abovevalue(or sellbids belowvalue) by bids at value isclearly
measurable. 28
Inanallpay auction,abidofzero isnotweaklydominated,andreplacing
bids above value by 0 is again clearly measurable. Using these two ideas, it is easy to
check that A10is satis ed forall the auctionsdiscussed inSection 3.3.
A useful observation is the following:
Lemma 5 Under A1-A10,given(e
i ;v
i );let b
i
be any bidvector suchthatp
ihe i (b ih )<v ih for someh e i , or p ihe i (b ih )>v ih for some h>e i . Then (e i ;v i ;b i )2= W i . 28
In anunfair rstprice auctionin which player1pays, say, 2/3of his bid, onewouldreplace bids
above3v
i
=2bybidsof3v
i =2:
Thatsuch abid vectoris weaklydominated at(e i ;v i );so that(e i ;v i ;b i )2= W i can be
seen as follows. If one raises a sell bid where p
ihe i (b ih ) <v ih
; then the only change can
be to either sell one less object, which originally sold at a loss, or to raise the market
price asaseller,eitherofwhichbene tstheplayer. ByA7,therecanbenochangeifone
was a net buyer before the change in bid. A similar argument applies when one lowers
a buy bid where p
ihe i (b ih ) > v ih
: Note that the same argument will be true for nearby
(e 0 i ;v 0 i ;b 0 i ): Hence, a neighborhood of (e i ;v i ;b i ) is outside of W 0 i ; and so (e i ;v i ;b i ) 2= W i :
A slightly more detailedproof appears in the appendix.
We say thatapro le ofdistributionalstrategies m isundominated if each m i places probabilityone on W i .
4 Invariance and Existence
We nowstate our rst main result.
Theorem 6 If an auction (P;o;t;b
0
) satis es A1-A10 and o is trade-maximizing, then
it has atleastoneundominated
equilibrium. Moreover,ifm issuchanequilibrium,then
the probability of competitive ties under m is 0; and m remains an equilibrium under
any omniscientand e ectivelytrade-maximizingtie-breakingrule,includingstandard
tie-breaking.
Our route for the proof of Theorem 6 is as follows. We begin with invariance: we
show thatany undominated
equilibriummust have zeroprobability ofany playerbeing
involved inacompetitivetieandwould alsobeanequilibriumifwechangedthe method
of tie-breaking to any other e ectively trade-maximizing omniscient tie-breaking rule.
Usinginvariance,ifwecanestablishexistenceofanequilibriuminnon-weaklydominated
strategies for some omniscient tie-breaking rule, this implies existence of (the same)
equilibrium under any tie-breaking rule, omniscient or standard. This second step is
fairly easily established viaeither of tworesults, either JSSZ, orReny.
The discussion of invariance appears in the next subsection (4.1). The step from
invariance to existence is in subsection (4.2), with additional details in the appendix.
Those not interested inthe proof can proceed directlyto Section5 to nd results
estab-lishing theexistence ofequilibriawith apositiveprobabilityof trade,animportantissue
The fact that ties are the critical worry for establishing existence of equilibrium follows
from the fact that ties are the only potential points of discontinuity. So, intuitively, if
we establish that players prefer to avoid ties, then we show that the discontinuities are
not important,which inturn allows us to establishthe existence of equilibrium. Let us
gorightto the heart of the matter.
Lemma 7 Considerany auction(P;o;t;b
0
) satisfyingA1-A10 and any pro le of
strate-gies m. For each " >0 and any bidder i2N there exists m 0 i within " of m i 29 such that m i ;m 0 i is tie-free for i 30 and i (m i ;m 0 i ) i (m i ;m i ) ".
Lemma 7 shows that for any pro le of strategies, any player can nd a close-by
strategy that doesnot involve any ties anddoesnearly aswellas heroriginal strategy. 31
Note that the Lemma 7 does not put any requirements on o or on m, and so it
allowsfortie-breaking that isnottrade-maximizingandforstrategies that areinweakly
dominated strategies.
Lemma 8 Fix an auction (P;o;t;b
0
) satisfying A1-A10. Let m be undominated
, and
either have a positive probability of ties where o is not trade-maximizing or a positive
probability of competitive ties. Then, there exists some bidder i 2 N and a strategy m 0 i such that m i ;m 0 i
is tie-free for i and
i (m i ;m 0 i ;P;o;t;b 0 )> i (m;P;o;t;b 0 ):
Lemma8shows thatforanypro leofstrategiesplacingprobabilityoneontheclosure
of the set of undominatedstrategies, but involvinga positiveprobability of competitive
tiesornon-competitiveonesthatarenottrade-maximizing,someplayerhasanimproving
deviation. This implies that if there exists an undominated
equilibrium, then it must
not involve any such ties.
The proofs of Lemmas 7 and 8 appear in the appendix. The idea behind Lemma 7
is fairly straightforward. Essentially, one bumps bids b
ih for which v ih > p ihe i (b ih ) up
slightly and bids for which v
ih p
ihe
i
(b) down slightly in such a way as to avoid bids
madebyotherplayers withpositiveprobability. Any changeintradethisbringsabout is
atmostslightlyunpro table. Forexample,if abuybid isbumped up, andwinsanextra
object, then the payment for the object is approximately p
ihe
i (b
ih
): Since the change in
29
Usethetopologyofweakconvergence.
30 That is,m i ;m 0 i
leadstoaprobability0ofibeinginvolvedinatie.
31
It is importantto remarkthat this isnotthe sameasestablishing better-reply-securityasde ned
by Reny [27]. Better-reply-security does not hold here, as we discuss in moredetail below. We are
notconsideringallpayo sthat maybereachedin theclosureofthegraphof thegame; onlyonesthat
a ected. The detailed proof is slightly more involved, because (a) it has to be checked
that one can always perform this perturbation consistently across di erent h, (b) one
needs toperformthis perturbationinameasurable fashionacrossbids andtypessothat
the composition of the original distributional strategy and the perturbation remains a
validdistributionalstrategy,and (c) thepossibilitythat odd tiebreakingmightresultin
a smallchange inanon-marginalbid a ecting whether ornot a marginalbid wins must
betaken account of.
To see Lemma8,assume that thereis apositiveprobabilityof a competitivetieor a
non-competitivebutnon-trademaximizingtieatb
Sincemisundominated
,byLemma
5, a player will not submit a buy bid b
ih = b where v ih < p ihei (b ih
) or a sell bid where
v ih >p ihe i (b ih
): Sincethe distributionof valuesis atomless,thereiszero probabilitythat
v ih =p ihe i (b
): Hence, almost allthe buyers atthat tie would strictly prefer to buy, and
almost all sellers would prefer to sell. But, no matter what the omniscient tie-breaking
rule, if there is apositiveprobability of acompetitivetieor anon-competitiveand
non-trade-maximizing tieat some b
, then at least one player who would bene t from trade
is sometimes \losing" the tie, and so strictly bene t by the deviation described above.
When we put Lemmas 7and 8 together, we end up with the following implication.
Theorem 9 (Invariance)Ifanauction(P;o;t;b
0
)satis esA1-A10,andanundominated
pro le of distributional strategies m is an equilibrium, then under m there is zero
prob-ability of a competitive tie or non-competitive ties where o is not trade-maximizing, and
m remains an equilibrium for (P;o 0
;t;b
0
) for any trade-maximizing tie-breaking rule o 0
.
Note that the conclusion that we can switch from o to o 0
and still have m be an
equilibrium, is not a direct implicationof Lemma 8. It may be that m is a equilibrium
undero,buto 0
wouldinducesomeplayertodeviatetoestablishanewtie. Thispossibility
is ruled out using Lemma 7,as the followingshort proof shows.
Proof of Theorem 9: The fact that m must be free of competitive ties and
non-competitive ties whereo is not trade-maximizingfollows directly from Lemma8. Let us
argue that m is alsoan equilibrium for (P;o 0
;t;b
0
). Given that any ties occurring with
positiveprobability underm must be non-competitiveand whereo is trade-maximizing,
m must lead to the same payo vector u under both o and o 0
. Now, suppose to the
contrary of the theorem that m is not an equilibriumunder o 0
. Then there exists i and
m 0 i such that i (m i ;m 0 i ;P;o 0 ;t;b 0 ) >u i . By Lemma 7 we can nd m 00 i which is tie-free
for i and such that
i (m i ;m 00 i ;P;o 0 ;t;b 0 ) >u i . Since m i ;m 00 i
is tie-free for i it follows
that i (m i ;m 00 i ;P;o;t;b 0 ) = i (m i ;m 00 i ;P;o 0 ;t;b 0 ) > u i
, contradicting the fact that m
is an equilibriumato.
The conclusions of Theorem 9 can fail if one ventures beyond private values. This
Theorem 9 establishes that any undominated
equilibrium can only involve
trade-maximizingnon-competitiveties,andwillremainanequilibriumforanytrade-maximizing
tie-breaking rule. Thus, to prove Theorem 6 we need only prove that there exists an
undominated
equilibriumfor some tie-breaking rule.
Wethinkthatitisinstructivetoo erproofsviabothJSSZand Reny,asatthis point
they are both fairly straightforward. Moreover, this clari es the relationship between
thesetwomethodologies,whichmaybeuseful infurther applicationsandin
understand-ing existence issues morebroadly.
4.2 Two Proofs of Theorem 3
We rstconstruct anauxiliarygamewheredominatedstrategies are penalizedaccording
to their distance from the set of undominated strategies. Equilibria in this game must
involve undominatedstrategies. It is also easyto see that these remain equilibria when
the penalty of domination is removed (being careful with some details regarding the
tie-breaking rule). This is stated inthe following Lemma.
Considerthe gameG(P;o;t;b
0
)inwhichwhenanyplayeriusesb
i with typee i ;v i ;he pays apenalty c i (e i ;v i ;b i
)in additionaltothe payo he receives from(P;o;t;b
0
), where
c
i
is the distance of a point fromthe set W
i .
32
Lemma 10 Consider an auction (P;o;t;b
0
) and the auxiliary G(P;o;t;b
0
) that
penal-izes weakly dominated strategies. Let m be an equilibrium of G(P;o;t;b
0
). Then, m is
undominated
and isan equilibrium of the original auction (P;o;t;b
0 ).
If G(P;o;t;b
0
) has an equilibrium m for some o, then by Lemma 10, so does the
original auction. Then by Theorem 9, m remains an equilibrium for any other
trade-maximizing o. This would then complete the proof of Theorem 6.
Proving Theorem 6 using JSSZ's Endogenous Tie-Breaking Rules.
Theorem 1inJSSZ impliesthat thereexists anequilibriumm inanaugmented form
of G(P;o;t;b
0
) where players also (truthfully) announce their types and tie-breaking
depends on those announcements. Those strategies remain an equilibrium when we
ignoretypeannouncements,and changethetie-breakingruletodirectlydependontypes
32
Takethedistanceofapointfromthe(closed)setW
i
tobetheminimumofthedistancesfromthat
